Monolithic multigrid for saddle point systems

Voronin, Alexey

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https://hdl.handle.net/2142/124245 Copy

Description

Title

Monolithic multigrid for saddle point systems

Author(s)

Voronin, Alexey

Issue Date

2024-04-10

Director of Research (if dissertation) or Advisor (if thesis)

Olson, Luke N

Doctoral Committee Chair(s)

Olson, Luke N

Committee Member(s)

• Gropp, William D
• Fischer, Paul
• MacLachlan, Scott

Department of Study

Computer Science

Discipline

Computer Science

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Multigrid
• Algebraic
• Geometric
• Iterative Methods
• Preconditioning
• Relaxation
• Additive Schwarz
• Vanka
• Stokes Equation
• High Order
• Finite Elements
• Saddle Point
• Systems
• Parallel
• MPI
• High Performance Computing

Abstract

In computational science and engineering, the discretization of coupled partial differential equations (PDEs) modeling multi-physics phenomena leads to large linear saddle-point systems. These systems encompass multiple interlinked unknowns, such as velocity, pressure, temperature, and charge, arising in simulations across domains like hydrocarbon extraction, biomedical engineering, and plasma physics. This dissertation focuses on developing a robust multigrid preconditioning framework for these systems using the Stokes equations as a representative model problem. A novel defect-correction approach is introduced for coupled systems, utilizing stable low-order re-discretizations to construct preconditioners for higher-order discretizations like Taylor-Hood and Scott-Vogelius elements. For Taylor-Hood, geometric multigrid performance is optimized through Local Fourier Analysis. Furthermore, a monolithic algebraic multigrid (AMG) method is developed, incorporating the defect-correction approach to robustly precondition these higher-order Stokes discretizations without relying on geometric information, unlike most existing approaches. To improve efficiency for high approximation orders, p-multigrid methods combining spatial and approximation order coarsening are presented. These methods outperform traditional spatial-only multigrid, especially for unstructured meshes. Different p-coarsening strategies are analyzed, and a robust approximate full-block factorization variant leveraging p-multigrid is introduced for the Scott-Vogelius discretization. Finally, patch relaxation techniques are proposed to reduce multigrid setup costs when solving sequences of related linear systems. This approach reuses patch factorizations between consecutive solves and updates only a subset of patches, minimizing overhead while maintaining fast convergence.

Graduation Semester

2024-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2024 Alexey Voronin

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